This martingale decomposition, together with the bound \(|\gamma _k| \le 2/v_n\) , appear in proofs of convergence to the Marchenko-Pastur distribution as \(\beta _n \rightarrow \beta > 0\)  [1]}, [2]}. Such proofs consider \(z\)  fixed. Here and in [3]}, [4]}, as \(v_n\)  may decay rapidly, a tighter bound on the moments of \(\gamma _k\)  is needed. Using the identity
\(\phantom{\,.} S_{nk} (S_{nk} - zI_{n-1})^{-1} = (S_{nk} - zI_{n-1})^{-1} S_{nk} = I_{n-1} + z (S_{nk} - zI_{n-1})^{-1}  \,,\) 
